Problem: Part of the graph of $f(x) = ax^3 + bx^2 + cx + d$ is shown. What is $b$?

[asy]
unitsize(1.5 cm);

real func(real x) {
  return((x + 1)*(x - 1)*(x - 2));
}

draw(graph(func,-1.1,1.5));
draw((-1.5,0)--(1.5,0),Arrows(6));
draw((0,-1)--(0,2.5),Arrows(6));

label("$x$", (1.5,0), E);
label("$f(x)$", (0,2.5), N);

dot("$(-1,0)$", (-1,0), SE, fontsize(10));
dot("$(1,0)$", (1,0), SW, fontsize(10));
dot("$(0,2)$", (0,2), NE, fontsize(10));
[/asy]
Solution: We have \[
0 = f(-1) = -a+b-c+d\]and \[0 = f(1) = a+b+c+d,
\]so $b+d=0$. Also $d=f(0) = 2$, so $b=\boxed{-2}$.